映射

Furthermore, in light of the φ-mapping theory the SU (2) Chern density is expressed as δ, and the zero points of 〓 field are characterized by the φ-mapping topological numbers—the Hopf indices and the Brouwer degrees.

进一步，利用Φ-映射理论作者证明SU（2）陈密度可表达成δ的形式，并阐明Φ场零点的拓扑特征可由Φ-映射拓扑数—Hopf指标和Brouwer度所表征。

In Chapter 2, we recapitulate some fundamental definitions and propositions of topological degrees and some fundamental results about the computation of topological degrees;in Chapter 3, applying relative definitions and propositions of topological degrees, we establish product formulae of topological degrees in product space, in particular, the product formula of fixed point index on product cone, moreover, combining with the fundamental propositions and the computation of fixed point index, we obtain some fixed point theorems on product cone.

为此，在第二章中，我们给出了拓扑度的基本概念和性质以及拓扑度计算方面的基本结果；在第三章中，运用拓扑度的有关知识，我们建立了乘积空间中各类映射的拓扑度的乘积公式，特别是乘积锥上的紧连续映射的不动点指数的乘积公式，进一步，结合不动点指数的基本性质和不动点指数计算方面的基本结果，得到了乘积锥上的若干不动点定理。

As applications, we prove the following results: If / has pointwise pseudo-orbit tracing property, for any k ∈ Z+, and fk is chain transitive, then for any k ∈ Z+, fk has open set transitive ; If f has pointwise pseudo-orbit tracing property, and for any n ∈ Z+,fn is chain transitive, then f has sensitive dependence on initial conditions; If f is open set mixing and has pointwise pseudo-orbit tracing property, then f has property P; Let f :→ be a homeomophism, then f is pointwise pseudo-orbit tracing property if and only if the shift map σf is pointwise pseudo-orbit tracing property.

作为应用，证明如下结论：若f具有逐点伪轨跟踪性质，且对任意k∈Z+，fk为链转换的，那么对任意k∈Z+，fk为开集转换；若f具有逐点伪轨跟踪性质，且对任意n∈Z+，fn为链转换的，则f具有初始敏感依赖性质；若f为开集混合的，且具有逐点伪轨跟踪性质，那么f具有性质P；设f：→是同胚映射，那么f具有逐点伪轨跟踪性质当且仅当移位映射σf具有逐点伪轨跟踪性质。

In chapter 4, basing theories and methods of scientific visualization, and artificial neural network BP algorithm, we integrate the Visual C++, OpenGL graphics library and Excel VBA technique to develop the program of artificial neural network and to make the BP algorithm visually, this program works can be divided into four parts: Using C language to develop program about BP algorithm; Using Visual C++, develop the GUI Interface, make input parameter visually; Using OpenGL graphic technique to display the training sample point in three dimension; at last using Excel DDE technique display the error graphic tables in Excel system In chapter 5, on the view of engineering application, we establish new method of surface reconstruction basing artificial neural network, develop interface program between module and commercial CAD/CAM system, meantime deeply discuss some key problems, for example, setting up the base plane, using the API technique, cutting and editing surface boundary, and also discuss the more compliant problem: how to intersect surface, at end we finish the work of translation from our surface reconstruction module to commercial CAD/CAM system, then make reverse engineering system basing artificial neural network more useful.

第四章基于科学计算可视化理论，依据人工神经网络BP算法理论模型，综合Visual C++，OpenGL图形库以及Excel VBA等多项软件开发技术，编制了人工神经网络程序，实现了BP算法的可视化映射。具体工作分为四部分：利用C语言实现人工神经网络BP算法；利用VisualC++的GUI技术开发图形用户界面，实现参数设置可视化；利用OpenGL图形技术进行三维映射，显示学习样本及训练样本点；利用微软电子表格DDE动态数据交换技术，在Excel上动态显示学习误差曲线图。第五章从工程应用的角度出发，提出了一种新的基于人工神经网络算法的曲面裁剪重构方法，完成了曲面重建模块与通用CAD/CAM系统的接口设计工作，对其中的若干关键问题进行了深入讨论，例如基平面设定、API技术的应用、边界裁剪等问题，同时，对曲面计算中较为困难的曲面相交问题也进行的专门探讨，最终完成了曲面重建模块向CAD/CAM系统的数据传输工作，使人工神经网络逆向工程系统趋向实用。

In Chapter 3, we pay our attention on Jordan triple derivable maps at zero point of nest subalgebras of factor von Neumann algebras.

证明了因子von Neumann代数中套子代数上零点Jordan三重可导映射是导子与恒等映射之和。

Different ways of mapping the IP cores to the underlying topology lead to different overall system performance and energy consumption. To find an optimal mapping of application to 2DMesh network, a model which describes the application must be found.

为此对2DMesh网络建立模型，寻找和设计高效的映射算法，使得片上网络在优化映射下充分发挥网络的并发计算资源，提高系统的性能，提高系统的实时性，降低系统的功耗。

Vehicle side slip angle is considered as mapping of time series of yaw rate and lateral acceleration. A uniform design project is used to select training samples, and the relationship of the three state parameters is established through neural network.

把汽车侧偏角看作横摆角速度和侧向加速度时间序列的映射，采用均匀设计方案对训练样本进行优选，通过神经网络建立三者之间的映射关系。

The unnaturalness of the mapping compels users to figure this relatship out anew every time they use the stove .

用户必须通过试验或者参考旋钮下的图标来确定，这种不自然的映射迫使用户在每次使用气炉时都不得不重新确定映射关系。

Users must find out by trial and error or by referring to the tiny icons next to the knobs. The unnaturalness of the mapping compels users to figure this relationship out anew every time they use the stove.

用户必须通过试验或者参考旋钮下的图标来确定，这种不自然的映射迫使用户在每次使用气炉时都不得不重新确定映射关系。

Vertex labeling is a mapping that maps the vertex set into integer set. On the other hand, edge labeling maps edge set into integer set.

一个图的顶点标号是图的顶点集到整数集的映射，而边标号则是图的边集到整数集的映射。

And Pharaoh spoke to Joseph, saying, Your father and your brothers have come to you.

47：5 法老对约瑟说，你父亲和你弟兄们到你这里来了。

Additionally, the approximate flattening of surface strip using lines linking midpoints on perpendicular lines between geodesic curves and the unconditional extreme value method are discussed.

提出了用测地线方程、曲面上两点间短程线来计算膜结构曲面测地线的方法，同时，采用测地线间垂线的中点连线和用无约束极值法进行空间条状曲面近似展开的分析。